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- Complete the lesson Assessment.
Lesson
The art however of discovering whether such numbers are commensurable by some number besides unity, or whether each of them is measured by unity alone, is as follows:
Two unequal numbers being given, it will be requisite to take the less from the greater, and if that which remains is still greater to take from it again the less, but if it is less to take it from the remaining greater number. And this must be done till unity at last prevents any farther subtraction, or till some odd number necessarily effects this, if both the proposed numbers are odd; and this number which remains will be the common measure of both.
Thus, for instance, let there be two numbers 9 and 29, of which it is proposed to investigate the common measure. From the greater let the less be taken, and there will remain 20. From this therefore let the less be again taken, and 11 will remain. From this again take 9, and 2 will remain. Let 2 then be taken from 9, and 7 will remain. From 7 let 2 be taken, and there will remain 5. Again, from 5 let 2 be taken, and 3 will remain. And farther still, if 2 be taken from 3, there will remain 1. And in the last place, if 1 be taken from 2, the subtraction will terminate in unity, which will alone be the measure of the two numbers 9 and 29. These numbers therefore, are called with reference to each other prime numbers.
Let it, however, be proposed to investigate the common measure of two other numbers 21 and 9, that it may be ascertained of what kind they are when compared with each other. Again therefore, from the greater number 21 let the less number 9 be taken, and 12 will remain. From 12 let 9 be taken, and there will remain 3. But if 3 be taken from 9, 6 will remain; from which if 3 be taken, 3 will be the remainder. And from this 3 cannot be taken so as to leave any remainder. Hence these numbers are commensurable, and 3 is their common measure.
Assessment
